Preheat the oven to 350°F (175°C).
Heat 1 tablespoon of the olive oil in a large skillet over medium-low heat.
Add the sliced mushrooms and sauté about 6 minutes until they release moisture and begin to brown.
Add the sliced red onion, minced garlic, and red bell pepper and continue sautéing about 4 minutes until the onion and pepper are tender.
Push the vegetables to the side and increase heat to medium; add the sirloin strips and cook 5–10 minutes, stirring occasionally, until no longer pink.
Season the mixture with dried oregano, black pepper, and salt to taste; stir well, cover, and simmer 5 minutes more. Drain any excess liquid.
Brush the remaining 1/2 tablespoon olive oil on both sides of the pita halves, place them on a cookie sheet, and warm in the oven 3–5 minutes.
Fill each warmed pita pocket with the meat-and-vegetable mixture, divide romaine lettuce among them, and top with a dollop of Greek yogurt or sour cream if using.